For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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99 views

Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
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1answer
77 views

Find the pdf of $Y = g(X)$, where $X$ is a uniform random variable

The question is as follows: Let $X$ be a uniform random variable over $(-1,2)$. Let $g(x) = |x|$. Find the pdf of $Y = g(X)$. And here is my take so far: $$f(x) = \begin{cases} 1/2 & \text{ ...
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1answer
49 views

Transformation of variables for a non-monotonic function

Question: Let $U \sim \mathrm{Unif}(−α, α)$ follow the uniform distribution on the interval $(−α, α)$ for some parameter $α > 0$ and consider the transformed random variable $X = \sin(U)$. ...
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0answers
28 views

Distribution of function of uniformly random variables

I am sorry if there is no simple answer to this or the answer is completely obvious but I am approaching my wits end here. Probability isn't my forte, nor am I even a mathematician. I am essentially ...
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1answer
237 views

Uniform Distribution with Independent Random Variables to compute mean of the present value of a bond.

John wants to purchase a bond which will pay him $X$ thousand dollars after two years, where $X$ is equally likely to be any of the numbers in the set $\{0, 1, 2, 3, 4, 5\}$. John believes that the ...
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1answer
30 views

Uniform Distribution - Change of Variable

I have been stuck on the following question If $X$ has a cumulative distribution $F(x)$, then show $Y = F(X)$ has a uniform distribution with $U(0,1)$. I attempted to solve this problem by first ...
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0answers
14 views

Uniformly distributed arrival times, each willing to wait 15 minutes, what is the probability they meet? [duplicate]

Alice and Bob agree to meet for lunch on a certain day at noon. However, neither is known for punctuality. They both arrive independently at uniformly distributed times between noon and 1 pm on that ...
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3answers
40 views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
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1answer
34 views

When is an improper Riemann integral equal to Lebesgue integral

My original problem is given $X_i\sim^{iid}U[0,1]$, find $$\lim_{n \rightarrow \infty} (X_1X_2 \cdots X_n)^{1/n} = \lim_{n \rightarrow \infty} (\prod_{i=1}^{n} X_i)^{1/n}$$ Well, $$\lim_{n ...
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1answer
50 views

Uniform distribution on $[0,1]$ and random variable $Y=\frac{U}{e^{1-U}}$

$U$~$Unif[0,1]$ and we have the random variable $Y=\frac{U}{e^{1-U}}$. Find the density function of $Y$. So far I have that $0\le Y\le1$ and that... $$F_Y(t)=P(\frac{U}{e^{1-U}}\le t)=P(\ln ...
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0answers
200 views

p-value of uniformity of given distributions,Matlab

Given a vector of real numbers $[a_0,...,a_n]$, how do I find the $p$-value (in Matlab, say) that it is drawn from the uniform distribution over [0,1]? I.e. $H_0$ is the hypotheses that it is drawn ...
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2answers
73 views

Probability Xavier and Yolanda meet for lunch

Xavier and Yolanda plan to meet for lunch between noon and 1 p.m. They arrive independently with uniform distribution on [0, 1]. Yolanda will wait 30 min. for Xavier, but Xavier will only wait 15 min. ...
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1answer
32 views

$P\left(X+\frac{10}{X}>7\right)$ of a uniform distribution

Problem: $X$ has a continuous uniform distribution on $[0,10]$. Find $P\left( X + \frac{ 10 } { X } >7\right)$. So far, I have the PDF $f(x) = 1/10$ and CDF $F(x) = x/10$ for $0 < x < 10$. ...
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1answer
35 views

Density function of uniform prob distribution

Let $X ∼\operatorname{Uniform}(0,1)$. Find the density function of $Y = e^X$. I got to: $F_Y(y)$=$P(Y\le y)$=$P(e^X\le y)$=$P(X\le \ln(y))$ Not sure where to go from here?
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1answer
41 views

Uniform distribution probability calculation

Here is an exam problem with the work shown: A man and a woman agree to meet at a certain location at about 12:30 pm. The man will arrive at a time uniformly distributed between 12:15 and 12:45, ...
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0answers
11 views

Partition Theorem to show $P(W \gt Z)$

I am confused as to how to use the partition theorem on the following example? Any help is appreciated! Suppose that W has a U(0,1) distribution and suppose that W is independent of the random ...
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0answers
52 views

approximate a probability distribution by moment matching

I have a 60-40 weighted distribution, of uniform(0,7.5) and uniform(7.5,10) respectively, i.e. $$f_X(x)=(0.6/7.5)1_{x∈[0,7.5)}+(0.4/2.5)1_{x∈[7.5,1]}$$ I have worked out that $$E(X) = 0.6(7.5/2) + ...
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1answer
64 views

Variance of a weighted uniform distribution

Given a weighted uniform distribution, where it is a 60-40 mixture of uniform(0,7.5) and uniform(7.5,10), I have found the mean to be $$E(X) = 0.6(7.5/2) + 0.4((10+7.5)/2)$$ How do I find the ...
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1answer
38 views

Multivariate transformation of three independent variables

An insurance company offers the following insurance package to a customer with three businesses on the same street. The insurer will pay for all damages to the business that incurs the most damage, ...
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0answers
22 views

Identify a probability distribution with coordinate transformation

I have a problem with this task: We have a random variable $X:\Omega \rightarrow \mathbb R^2$, which is uniformly distributed on $K:= \{(x_1,x_2) \in \mathbb R^2 : \sqrt{x_1^2+x_2^2} \le 1 \}$ Now I ...
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2answers
76 views

PDF of several draws from an uniform distribution?

Suppose I draw several times from an uniform distribution, $X\sim\mathcal{U}(0, 1]$. (I'll use $\mathrm{R}()$ to denote an independent drawing.) What is then the PDF of several draws, added and/or ...
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1answer
21 views

$X$ RV with cdf $F$, $W \sim U[0,1]$ independent $\Rightarrow$ $V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1]$

I try to prove: Let $X$ be a discrete random variable with cdf $F$, $F_{-}(x):=P(X<x)$, $W \sim U[0,1]$ a random variable and $X, W$ independent. Then $$V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1].$$ My ...
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0answers
16 views

Mapping Function for Non-Uniform Circular Distributions

Note: I a programmer, not a mathematician. For a non-uniform distribution of points that occur on the unit circle is there a function or mapping that returns the k$th$ point in this distribution? For ...
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0answers
37 views

Uniform distribution over $\mathbb{R}^2$

Suppose, on $\mathbb{R}^2$, that $X$ is a random variable which takes values uniformly at random over the $\textit{line segment}$ from $(0,0)$ to $(a,a)$, where $a > 0$ is a positive constant. How ...
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0answers
38 views

poisson and uniform distributions

I have an answer to this question from someone else but I do not think it is right. Here is the question: Customers arrive at a bank at a Poisson rate lambda. Suppose two customers arrive during the ...
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1answer
32 views

Compound of uniform and gamma probability distributions

I am trying to compute the distribution of a uniform distribution whose upper limit is drawn from a gamma distribution. That is, $X \sim \Gamma(\alpha,\beta)$ $Y \sim U(0,X)$ We know: ...
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1answer
31 views

MLE of uniform distibution again

I've struggled for hours with a seemingly simple problem, I'm supposed to compute the MLE for $\theta$. We have $(y_1, y_2...y_n)$ obervations with a uniform distribution. The density function is as ...
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1answer
27 views

Probability of A winning given a uniform distribution

If the interval of A has been uniformly chosen as [0,1] and B as [0,6] then what is the probability of A being a lower number than B? I'm completely lost here, do I somehow calculate the uniform ...
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0answers
15 views

How do I normalize a uniform dist?

If I have a uniform distribution over A to B, and I want to find the prb of a trial being within 1 std dev, once I have the mean and std dev, how do I normalize this, so the mean is 0 and a std dev is ...
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1answer
47 views

Meaning of probability density function - continuous random variables

Suppose we have a random variable X uniformly distributed over the interval (0,1). The probability density function of X is given by: $$f(x)=\left\{\begin{array}{l} 1 \space\space if \space\space ...
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1answer
24 views

Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?

How to show that $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent, if $X_1$ and $X_2$ are i.i.d. exponential with mean $1$? Is there a simple way to see this?
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2answers
69 views

Uniform Probability Distribution CDF and Probability

Suppose a value $x$ is chosen at random in the interval $[0,10]$. In other words, $x$ is an observed value of a random variable $X \sim \mathrm{UNIF}(0,10)$. The value $x$ divides the interval ...
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1answer
92 views

Prove the median of a uniform distributions is $\frac{1}{2}(a+b)$

Let X~U(a,b) with a and b in the real line, such that b>a with X's pdf given by $$f_X(x)=\frac{1}{b-a}\mathbb{1}(a<x< b)$$ Show the median of X's distribution is given by $$m=\frac{1}{2}(b+a)$$ ...
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1answer
41 views

Expectation of Uniform Distribution with Sin

Let $X ∼ \operatorname{Unif} (a, b)$. What is $E[\sin(X)]$? I know how to find the expectation of a uniform distribution, but I'm unsure how to find $E[\sin(x)]$. ...
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1answer
68 views

Height of uniform distribution?

If X follows a uniform distribution in the interval [2, 7], what is the height of the probability density function (pdf) at x = 4? I'm new to Probability & Statistics and will appreciate any ...
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0answers
56 views

Find the distribution of $|X-Y|$ if $X$ and $Y$ are i.i.d. uniform on $[0,1]$

$X$ and $Y$ are independent random variables uniformly distributed over $[0,1]$. I want to find the CDF of $|X-Y|$. I could use convolution but I wan't to calculate this more "directly". Here is my ...
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1answer
74 views

Construct random variable from uniform distribution [closed]

I am trying to do this problem: Suppose $ U$ is a random variable with distribution $\mathcal U(0,1)$. Find a function $g$ such that $g(U)$ has distribution: i)$\mathcal E(1)$ ii) ...
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1answer
21 views

Average distance to the closest neighbor

Suppose that 3-dimensional space contains countably infinite number of randomly positioned particles (points), on average $1$ particle per a unit of volume. Their distribution is homogeneous and ...
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2answers
124 views

Probability of waiting before meeting in case of two uniformly distributed random variables

I have a question that goes like this. Two people, X and Y, decide to meet at a particular time. The probability of them both being late is uniformly distributed between 0 and 60. Person X is always ...
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1answer
52 views

How can I find a minimal sufficient statistic for θ from a U(θ-1, θ+1)?

Suppose X1:n is a random sample from a U(θ-1, θ+1) distribution. Find a minimal sufficient statistic for θ. Show that the MLE of θ is not well defined. Suggest an alternative “sensible” point ...
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0answers
37 views

convergence in distribution

Let $U_{t}$ be iid Uniformly distributed on (0,1). Suppose $\hat{\theta}_{T}\stackrel{d}\rightarrow \theta^{*}$ with $\theta^{*}$ some random variable on (0,1). I believe $\sum_{t=1}^{T}I(U_{t}\leq ...
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0answers
37 views

Prove that uniform distribution on a set of vertices $V$ is stationary if the graph is regular.

I was going through Random walks on graphs: A survey It was stated that: Uniform distribution on a set of vertices $V$ is stationary if the graph is regular. Can anyone give me some hints to ...
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1answer
45 views

PDF for $\frac1a Uniform(−a,a)$

Problem: Let $X$ be $Uniform(−a,a)$ distrubuted. Calculate the PDF for $Z = \frac1a abs(X)$. Attempt: I think graphically here. $X$ is $U(-a,a)$ with $PDF = \frac1{2a}$ so $abs(X)$ is $U(0,a)$ with ...
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1answer
58 views

Statistics find E(x), the number of distinct elements in uniformly distributed pool of items

Question: Suppose there are Y types of balls in a bucket, which are normally distributed and independent. Hence the probability of picking one type out is $\frac{1}{Y}$. Let $x$ be the number of ...
2
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1answer
81 views

Discrete conditional probability and expectation

I'm having troubles in solving this probabilty problem. A group of $n$ players is given; they are divided into two teams with the following procedure. A number $X$ is chosen randomly from the set ...
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2answers
24 views

How can we find the distribution function of an Uniform Random variable with Random variable bounds?

X is a uniform random variable in (0,1) and Y is a uniform random variable in (X,1). How can I find the probability density function of Y? I thought and searched a lot and I found nothing. please help ...
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0answers
45 views

Expected value of a biased coin

I came across this question of finding expected value of p given Head showed up in last toss of coin. Here p is probability of a biased coin showing H and p is given as uniformly distributed.
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1answer
33 views

Calculation of quantiles of a uniform distribution over a sphere

How do we calculate quantiles of a uniform distribution over a sphere ? Can anyone provide me with a tutorial ?
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1answer
73 views

CDF of Function of a Random Variable

This question is pretty basic but I am still having trouble understanding it. This is the question, verbatim, from my textbook: "If $X$ is a random variable that is uniformly distributed between ...
2
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1answer
51 views

Variance of unbiased estimator of a random sample from the uniform distribution?

Let X1,...,Xn be a random sample from the uniform distribution on the interval from 0 to theta for some theta>0. I want to find the variance of the unbiased estimator. I know the unbiased estimator ...